Delocalization and quantum diffusion of random band matrices in high dimensions I: Self-energy renormalization

Fan Yang University of Pennsylvania Horng-Tzer Yau Harvard University Jun Yin University of California, Los Angeles

Probability mathscidoc:2110.28001

2021.10
We consider Hermitian random band matrices $H=(h_{xy})$ on the $d$-dimensional lattice $(\Z/L\Z)^d$. The entries $h_{xy}$ are independent (up to Hermitian conditions) centered complex Gaussian random variables with variances $s_{xy}=\mathbb E|h_{xy}|^2$. The variance matrix $S=(s_{xy})$ has a banded structure so that $s_{xy}$ is negligible if $|x-y|$ exceeds the band width $W$. In dimensions $d\ge 8$, we prove that, as long as $W\ge L^\e$ for a small constant $\e>0$, with high probability most bulk eigenvectors of $H$ are delocalized in the sense that their localization lengths are comparable to $L$. Denote by $G(z)=(H-z)^{-1}$ the Green's function of $H$. For $\im z\gg W^2/L^2$, we also prove a widely used criterion in physics for quantum diffusion of this model, namely, the leading term in the Fourier transform of $\E|G_{xy}(z)|^2$ with respect to $x-y$ is of the form $(\im z + a(p))^{-1}$ for some $a(p)$ quadratic in $p$, where $p$ is the Fourier variable. Our method is based on an expansion of $T_{xy}=|m|^2 \sum_{\alpha}s_{x\alpha}|G_{\alpha y}|^2$ and it requires a self-energy renormalization up to error $W^{-K}$ for any large constant $K$ independent of $W$ and $L$. We expect that this method can be extended to non-Gaussian band matrices.
Random band matrices, delocalization, quantum diffusion
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@inproceedings{fan2021delocalization,
  title={Delocalization and quantum diffusion of random band matrices in high dimensions I: Self-energy renormalization},
  author={Fan Yang, Horng-Tzer Yau, and Jun Yin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20211001121521721866872},
  year={2021},
}
Fan Yang, Horng-Tzer Yau, and Jun Yin. Delocalization and quantum diffusion of random band matrices in high dimensions I: Self-energy renormalization. 2021. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20211001121521721866872.
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